# 06-07 Ladder slides down the wall

**Problem 06**

A ladder 20 ft long leans against a vertical wall. If the top slides downward at the rate of 2 ft/sec, find how fast the lower end is moving when it is 16 ft from the wall.

**Solution 06**

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$2x \dfrac{dx}{dt} + 2y \dfrac{dy}{dt} = 0$

$x \dfrac{dx}{dt} + y \dfrac{dy}{dt} = 0$

$x \dfrac{dx}{dt} + y (-2) = 0$

$x \dfrac{dx}{dt} - 2y = 0$

when x = 16 ft

$16^2 + y^2 = 20^2$

$y = 12 \,\, \text{ ft }$

$16 \dfrac{dx}{dt} - 2(12) = 0$

$\dfrac{dx}{dt} = 1.5 \, \text{ ft/sec}$ *answer*

**Problem 7**

In Problem 6, find the rate of change of the slope of the ladder.

**Solution 07**

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$m = \dfrac{y}{x}$

$\dfrac{dm}{dt} = \dfrac{x \dfrac{dy}{dt} - y \dfrac{dx}{dt}}{x^2}$

From Problem 6:

y = 12 ft

dx/dt = 1.5 ft/sec

dy/dt = -2 ft/sec

$\dfrac{dm}{dt} = \dfrac{16(-2) - 12(1.5)}{16^2}$

$\dfrac{dm}{dt} = \dfrac{-50}{256}$

$\dfrac{dm}{dt} = -\dfrac{25}{128} \,\, \text{ per second } \,\, $

$\dfrac{dm}{dt} = \dfrac{25}{128} \, \text{ per second decreasing}$ *answer*

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